Phonon sidebands of color centers in hexagonal boron nitride

Low temperature photoluminescence spectra of the phonon sidebands of a color center in hexagonal boron nitride are compared to an independent boson model. We infer that the LA-phonon sideband is described by deformation coupling proportional to in-plane strain, resulting in a phonon bath that is effectively two-dimensional. For optical phonons, sharp resonances close to turning points in the phonon dispersion are observed. We infer that the TO-electron coupling is described by deformation coupling proportional to in-plane lattice displacement, resulting in a TO bath that is also effectively two-dimensional. By contrast, for the LO branches we infer that the Fröhlich coupling depends on the interactions between adjacent layers, and propose that the 200-meV peak is a signature of a few-layer sample. This work highlights features of electron-phonon coupling that arise from the layered structure of h-BN.

Figure 1

Comparison of PL at 20 K and 275 K for emitter I. The ZPL can be identified by its large drop in intensity with temperature. By comparison to the phonon dispersion curves [21] the acoustic and optical sidebands can be identified. The markers indicate data points, and lines are fits to data from Ref. [21]. For the $\mathrm{LO}\left({\mathrm{E}}_{1\mathrm{u}}\right)$ band, a fit to ab initio calculations in Ref. [21] is used. (Inset) 2D Brillouin zone.

Figure 2

Closeup of acoustic phonon sideband. Note that sideband decays exponentially with detuning. Lines are calculations that assume electron phonon spectral densities, $J\left(\omega \right)\sim {\omega }^n{f}^2\left(\omega \right)$ of order $n=1$ (cyan), 2 (blue), 3 (green). More specifically, the case $n=d$ where a deformation coupling with $f\left(0\right)=1$ is assumed. These calculations use a linear dispersion. Qualitatively, the $n=2$ case fits best. The red line shows the case $n=2$, using Eq. (3) to estimate the anisotropy of the nonlinear dispersion curves, and gives a slightly better fit to data. The parameters used are as follows: ${\mathcal{D}}_{1D}=4\mathrm{eV},{\rho }_{1D}=1.65\times {10}^{-16}\mathrm{kg}{\mathrm{m}}^{-1},{\sigma }_{1D}=0.21\mathrm{nm}$; ${\mathcal{D}}_{2D}=10.5\mathrm{eV},{\rho }_{2D}=0.76\mathrm{mg}{\mathrm{m}}^{-2},{\sigma }_{2D}=0.35\mathrm{nm}$; ${\mathcal{D}}_{3D}=26\mathrm{eV},{\rho }_{3D}=2.18\times {10}^3\mathrm{kg}{\mathrm{m}}^{-3},{\sigma }_{3D}=0.49\mathrm{nm}$.

Figure 3

(a) Dispersion curves of the optical phonon branches. The markers are data, and the lines are fits to data taken from Serrano et al. [21]. The $\mathrm{LO}\left(E_{1u}\right)$ band is a fit to ab initio calculation of Ref. [21]. Vertical lines are aligned to turning points in the dispersion curves where the density of states is high. Comparison of data to calculations considering a single band: (b) $\mathrm{LO}\left(E_{1u}\right)$, (c) $\mathrm{LO}\left(E_{2g}\right)$, (d) TO. We consider a Fröhlich (blue, cyan) and deformation (red, orange) coupling with (blue, ${\epsilon }_{\mathrm{eff}}=22.2,\sigma =0$); (cyan, ${\epsilon }_{\mathrm{eff}}=33.3,\sigma =0.2\mathrm{nm}$);(red, $\mathcal{M}=72\mathrm{eV}{\mathrm{nm}}^{-1}$); (orange, $\mathcal{M}=341.5\mathrm{eV}{\mathrm{nm}}^{-1}$). For the TO branch only the deformation coupling is presented. (e) A fit to data assuming Fröhlich coupling to two LO branches, and deformation coupling to the TO. Values of ${\epsilon }_{\mathrm{eff}}\left(\mathrm{LO}\left(E_{1u}\right)\right)=33.3,{\epsilon }_{\mathrm{eff}}\left(\mathrm{LO}\left(E_{2g}\right)\right)=66.7,{\sigma }_{\mathrm{LO}}=0.2\mathrm{nm}$, ${\mathcal{M}}_{\mathrm{TO}}=153\mathrm{eV}{\mathrm{nm}}^{-1},{\sigma }_{\mathrm{TO}}=0.1\mathrm{nm}$ are used.

Figure 4

Photoluminescence spectra of emitter II. (Top) Dispersion curves of bulk-phonon modes. The markers are data points, and the lines fit to data or ab initio calculations of Ref. [21]. The $z$-polarized ZA and ZO branches are shifted to the blue by 5.5 meV to match peak A. The peaks A–O are identified in Table 2. To aid comparison a number of construction lines aligned with peaks in PL have been drawn.